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Chapter 5: Problem 19

Evaluate the sums. $$ \text { a. }sum_{k=1}^{10} k \quad \text { b. } \sum_{k=1}^{10} k^{2} \quad \text { c. } \sum_{k=1}^{10} k^{3} $$

Short Answer

Expert verified
a) 55, b) 385, c) 3025.

Step by step solution

01

Understanding the Problem

We need to evaluate three different sums involving powers of integers from 1 to 10. These are: a) sum of integers, b) sum of squares of integers, and c) sum of cubes of integers.
02

Solve Part a

The sum of the first 10 positive integers can be calculated using the formula for the sum of an arithmetic series: \[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]. Applying this to \( n = 10 \), we get: \[ \sum_{k=1}^{10} k = \frac{10(10+1)}{2} = \frac{10 \times 11}{2} = 55 \].
03

Solve Part b

The sum of the squares of the first 10 positive integers can be calculated using the formula: \[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \]. Applying this formula for \( n = 10 \), yields: \[ \sum_{k=1}^{10} k^2 = \frac{10(10+1)(2 \times 10 + 1)}{6} = \frac{10 \times 11 \times 21}{6} = 385 \].
04

Solve Part c

The sum of the cubes of the first 10 positive integers is given by the formula: \[ \sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 \]. For \( n = 10 \), we calculate: \[ \sum_{k=1}^{10} k^3 = \left(\frac{10 \times 11}{2}\right)^2 = 55^2 = 3025 \].
05

Summarize Results

We have evaluated each part of the sums separately. For part a, the sum is 55; for part b, it is 385; and for part c, the sum is 3025.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Series
An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. The series can be written as \( a, a+d, a+2d, a+3d, \ldots \), where \( a \) is the first term and \( d \) is the common difference.

The sum of the first \( n \) terms of an arithmetic series can be calculated using the formula:
  • \( S_n = \frac{n}{2} (2a + (n-1)d) \)
However, when the first term \( a \) is 1 and the difference \( d \) is also 1, this formula simplifies to:
  • \( S_n = \frac{n(n+1)}{2} \)
For example, to find the sum of integers from 1 to 10, you plug into the simplified formula:
  • \( S_{10} = \frac{10 imes 11}{2} = 55 \)
This method is both efficient and easy to use for rapidly finding the sum of an arithmetic sequence.
Sum of Squares
The sum of squares is concerned with adding together the squares of a series of numbers, such as \( 1^2, 2^2, 3^2, \ldots, n^2 \). This measure is crucial in various mathematical contexts, such as statistics and physics.

The formula to calculate the sum of squares of the first \( n \) natural numbers is:
  • \( \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \)
Applying the formula to the first 10 natural numbers, we find:
  • \( \sum_{k=1}^{10} k^2 = \frac{10(10+1)(2 \times 10 + 1)}{6} = 385 \)
The sum of squares plays a key role in computations involving variances and standard deviations, making understanding it all the more essential.
Sum of Cubes
The sum of cubes is the total of each number cubed from 1 to \( n \). This concept is often used in algebra and number theory to discover interesting properties of numbers.

The formula to determine the sum of cubes is an elegant relation:
  • \( \sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 \)
In simpler terms, it's the square of the sum of the first \( n \) natural numbers.
  • For example, \( \sum_{k=1}^{10} k^3 = 55^2 = 3025 \)
This result is quite fascinating as it suggests a powerful geometric pattern between simple sums and their cubic counterparts. Understanding the sum of cubes deepens one's grasp of mathematical symmetry and unity.

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Most popular questions from this chapter

Solve the initial value problems in Exercises \(55-60 .\) $$\frac{d^{2} y}{d x^{2}}=4 \sec ^{2} 2 x \tan 2 x, \quad y^{\prime}(0)=4, \quad y(0)=-1$$

Evaluate the integrals in Exercises \(17-50\) $$ \int \sqrt{\frac{x^{4}}{x^{3}-1}} d x $$

Find the areas of the regions enclosed by the lines and curves. $$ y=x^{4}-4 x^{2}+4 \text { and } y=x^{2} $$

Evaluate the integrals in Exercises \(17-50\) $$ \int \frac{1}{\theta^{2}} \sin \frac{1}{\theta} \cos \frac{1}{\theta} d \theta $$

Find the areas of the regions enclosed by the lines and curves. $$ x=y^{3}-y^{2} \quad \text { and } \quad x=2 y $$
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